# Duoprism (EntityClass, 13)

### From Hi.gher. Space

A **duoprism** is the Cartesian product of two polygons. In other words, it is the set of all combinations of points (w,x,y,z) where (w,x) is in the first polygon, and (y,z) is in the second.

A duoprism made from the Cartesian product of an *m*-polygon and an *n*-polygon is referred to as an *m*,*n*-duoprism or an *m*-gonal *n*-gonal duoprism (George Olshevsky).

An *m*,*n*-duoprism is bounded by *m* *n*-gonal prisms and *n* *m*-gonal prisms, for a total of *m+n* cells. The two types of cells lie on two mutually orthogonal planes. Cells within a ring are connected to each other via their *m*-gonal (or *n*-gonal) faces. Cells from different rings are joined to each other via their square faces.

## Special cases

The duoprism is convex if the two source polygons are convex.

The two polygons are usually assumed to be regular. If the edge lengths of the two polygons are equal, the resulting duoprism will be a uniform polychoron. (Note that the regularity of the polygons by itself is not sufficient to make the duoprism uniform.)

A 4,4-duoprism is equivalent to the tesseract.

## Limiting shapes

As *m* approaches infinity, an *m*,*n*-duoprism approaches an *n*-prismic cylinder (i. e., the Cartesian product of an *n*-polygon with a circle). The 4-prismic cylinder is equal to the cubinder.

If *n* is also allowed to diverge to infinity, the limiting shape is the duocylinder (i. e., the Cartesian product of two circles).

Due to this property, duoprisms serve as good non-quadric approximations of the prismic cylinders and the duocylinder. This is useful for implementing programs for rendering these objects, because it is much easier to implement a renderer based on polygons (using a large but finite value for *m* and/or *n*) than one based on 4D quadric surfaces.

## Related shapes

The double-ring structure of duoprisms is related to the Hopf fibration of the 3-sphere, and thus shares similarities with all 4D shapes that approximate the 3-sphere.

One such shape where the double-ring structure is obvious is the grand antiprism, which consists of two rings of 20 pentagonal antiprisms (10 in each ring), with tetrahedra joining the two rings together. The grand antiprism can be constructed from the 600-cell by removing 20 vertices that lie along two mutually orthogonal rings and then recomputing the convex hull.